JournalsggdVol. 9, No. 4pp. 1153–1184

Boundary values, random walks, and p\ell^p-cohomology in degree one

  • Antoine Gournay

    Université de Neuchâtel, Switzerland
Boundary values, random walks, and $\ell^p$-cohomology in degree one cover
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Abstract

The vanishing of reduced 2\ell^2-cohomology for amenable groups can be traced to the work of Cheeger and Gromov in [10]. The subject matter here is reduced p\ell^p-cohomology for p]1,[p \in ]1,\infty[, particularly its vanishing. Results for the triviality of pH1(G)\underline {\ell^pH}^1(G) are obtained, for example: when p]1,2]p \in ]1,2] and GG is amenable; when p]1,[p \in ]1,\infty[ and GG is Liouville (e.g. of intermediate growth).

This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced p\ell^p-cohomology is equivalent to the absence of non-constant harmonic functions with gradient in q\ell^q (qq depends on the profile). In particular, one reduces questions of non-linear analysis (pp-harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).

Cite this article

Antoine Gournay, Boundary values, random walks, and p\ell^p-cohomology in degree one. Groups Geom. Dyn. 9 (2015), no. 4, pp. 1153–1184

DOI 10.4171/GGD/337